1. Timing of major earthquakes
In the last set of exercises, you looked at *the power* of earthquakes. Now, you will look the *timing* of earthquakes.
2. Models for earthquake timing
While the study of earthquake timing has produced many models, there are two main textbook models, which are the two we will study here. First, there is the Exponential model, which assumes that earthquakes happen like a Poisson process. This means that when the last earthquake happened has no bearing on when the next will happen. Then there is the Gaussian, or Normal, model. Under this model, earthquakes happen periodically.
3. Stable continental region earthquakes
As an example, let's look at the timing of earthquakes that happen away from faults in the so-called stable continental regions around the world. Here is the ECDF of the time between these quakes. I have omitted inter-earthquake times less than two weeks so as not to count aftershocks.
The time between earthquakes is clearly Exponentially distributed, with the theoretical distribution overlayed in green. This makes sense; occurrence of earthquakes around the world are probably not correlated.
The picture is not so clear, though, if we restrict ourselves to predicting the earthquakes we are more interested in: very powerful earthquakes along faults.
4. The Nankai Trough
For example, let's consider the magnitude eight-plus megathrust earthquakes that have happened along the Nankai Trough off the coast of Honshu in Japan.
5. Earthquakes in the Nankai Trough
In looking at the dates of the earthquakes, they seem to happen roughly every 200 years.
6. ECDF of time between Nankai quakes
Plotting the ECDF unfortunately does not immediately add all that much clarity. There simply are not many data points.
It is worth pausing to think about our representation of ECDFs. With these few data points, it is kind of hard to see the shape of the ECDF plotted as points.
7. Formal ECDFs
Formally, the ECDF at *x* is defined as the fraction of data points that are less than or equal to *x*. This is defined at all positions along the *x* axis, not just those corresponding to measured data points.
8. Formal ECDFs
So, formally, we should plot the ECDF like this.
9. Formal ECDFs
The correspondence between the two representations is clear with the dot representation you're used to overlayed in red.
10. Formal ECDFs
The function `dcst.ecdf()` will allow you to make plots of these formal ECDFs via the `formal` keyword argument.
It is a matter of opinion as to how you display your ECDFs. Either way has the same information. My personal preference is for dots for any dataset with more than 20 data points because the values of the measurements are very clear. But formal ECDFs are also always ok.
Now, what if you wanted to compare this ECDF to theoretical CDFs of the models for earthquake occurrence?
11. Generating theoretical distributions
You first compute the mean and standard deviation from the data, since these will be the best estimates for the parameters of the theoretical distributions.
To get the theoretical CDFs, you can use NumPy's `random` module to draw lots of samples out of the theoretical distributions, in this case Exponential and Normal. You can then use these to plot the theoretical CDFs.
12. Model for Nankai Trough
Performing those operations gives this result. The timing of the Nankai megathrust earthquakes seems to follow the Gaussian model more closely than the Exponential.
13. Let's practice!
Now it is your turn to take a look at the sequence of big earthquakes around Parkfield in recent years. Recent geologically speaking that is!